arXiv:nucl-th/0406076v1 24 Jun 2004
Dipole states in stable and unstable nuclei
D. Sarchi∗, P.F. Bortignon and G. Colò
Dipartimento di Fisica, Università degli Studi
and INFN, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy
May 6, 2017
Abstract
A nuclear structure model based on linear response theory (i.e.,
Random Phase Approximation) and which includes pairing correlations and anharmonicities (coupling with collective vibrations), has
been implemented in such a way that it can be applied on the same
footing to magic as well as open-shell nuclei. As applications, we have
chosen to study the dipole excitations both in well-known, stable isotopes like 208 Pb and 120 Sn as well as in the neutron-rich, unstable
132 Sn nucleus, by addressing in the latter case the question about the
nature of the low-lying strength. Our results suggest that the model
is reliable and predicts in all cases low-lying strength of non collective
nature.
PACS numbers: 24.30.Cz, 21.60.Jz, 27.60.+j
Giant Resonances (GR) in atomic nuclei lie at excitation energies above
the nucleon separation threshold (∼8-10 MeV), have different multipolarity
and carry different spin-isospin quantum numbers. They are associated with
the elastic, short-time (h̄/10 MeV ≈ 10−22 s) response of the nuclear system.
They have been observed throughout the mass table with large cross sections,
Present address: Institute of Theoretical Physics, CH 1015 Lausanne-EPFL, Switzerland.
∗
1
close to the maximum allowed by sum rule arguments, implying that a large
number of nucleons participate in a very collective nuclear motion [1, 2].
The giant dipole resonance (GDR) has been the first to be discovered, and
is probably the most studied among the various nuclear collective vibrations.
In well-bound systems it is often depicted as a coherent oscillations of essentially all protons against all neutrons, boosted by an initial displacement of
the centers of the two distributions induced by an external electromagnetic
field. The energy of this resonance in MeV, is given by the empirical 80 A−1/3
law.
In the neutron-rich isotopes, it has been suggested that part of the energy
provided by the external field can be used to excite a different kind of motion. In fact, if the valence neutrons occupy orbitals with lower energies and
larger radii than the inner, or ”core”, particles, these orbitals must be somehow decoupled from the others. Consequently, the excitations of the valence
neutrons should result in vibrations of these particles against the inner ones,
namely in modes with different characteristics than the standard GDR.
Recently [3] a systematic study in the neutron-rich isotopes 18−22 O, via
relativistic Coulomb excitation, has identified sizeable strength (about 10%
of the Thomas-Reiche-Kuhn (TRK) sum rule) below 15 MeV, that is, well
below the GDR region. The question about the collectivity of this strength
is left to theorists. So far, every microscopic model (either within the nonrelativistic [4] or relativistic [5] framework) seems to find only non-collective
states in the low-lying region. A further natural question which arises, is
whether this low-lying dipole strength systematically exists in the mediumheavy nuclei and whether a certain degree of collectivity can develop, with
increasing mass number. Another issue is whether the properties of the lowlying dipole can shed light on the (possible) modifications of the nuclear
effective Hamiltonian(s) when one goes far from the line of stability.
These questions have been attacked within the most widely used mean
field models [6], which however do not take into account the spreading effects,
as we will do in this work.
In fact, in nuclei the mean field defines a surface which can vibrate leading
to a spectrum of collective low-lying excitations (phonons). The coupling of
the nucleons to these dynamical vibrations, a process which goes beyond
mean-field, can strongly renormalize the single-particle motion by changing
the energy and the occupancy of the levels around the Fermi energy and,
eventually, by providing a single-particle spreading width Γ↓s.p. . In keeping
with the fact that the giant resonances can be viewed as correlated particle2
hole (p-h) or two quasi-particle (2qp) states, the very same mechanism will
produce their spreading width Γ↓GR , although quantitatively this width is
reduced compared to the sum of the particle and hole widths since this is
the strong effect of the coherence between the particle and hole motions as
shown in Ref. [7].
The aim of this paper is the following. If on one hand mean field studies are nowadays performed using sophisticated forces well linked to nuclear
matter properties, they do not include the phonon coupling mentioned above.
We would like to provide a unified description of the properties of the giant
resonances within a self-consistent model which includes such a coupling in
both magic and open-shell nuclei. In this sense, the work is a continuation
of the old pioneering calculations of Ref. [8], with the very clear improvement coming from the self-consistent use of an effective nucleon-nucleon interaction. Among other limitations, in Ref. [8] pairing was considered in
a simplistic way and it was not possibile to calculate the giant resonances
along an isotopic chain. As far as the particle-hole channel is concerned, a
consistent calculation of the dipole states was performed in Ref. [4]. In this
work, a Skyrme-type interaction was employed and the residual interaction
between quasiparticles was self-consistently derived. However, the pairing
contribution was dropped from the residual interaction. Nowadays, since extrapolations to neutron-rich nuclei are needed, it is desirable to be able to
reproduce the lineshape of the giant resonances in different nuclei without ad
hoc adjustments, neither in the particle-hole nor in the pairing channel. Thus,
we have accomplished the implementation a more general model compared
to that of [4], in the sense that we now treat consistently also the pairing
interaction. We test this new model, that we call QRPA-PC (Quasi-particle
Random Phase Approximation plus Phonon Coupling), in the case of dipole.
We analyze the well-known 208 Pb nucleus already, and also very recently [9],
investigated in many aspects, and also another stable, yet open shell nucleus,
namely 120 Sn. In both cases we can compare with measured photoabsorbtion
cross sections. Then, we make predictions for the neutron-rich isotope 132 Sn,
for which experimental results should be soon available.
Our starting point is a mean field calculation performed in the HF-BCS
(Hartree-Fock-Baarden-Cooper-Scrieffer) plus QRPA approach. As already
mentioned, the model is self-consistent, in the sense that a Skyrme force is
used in the particle-hole channel, while in the particle-particle one a zero-
3
range pairing interaction
Vpair (~x1 , ~x2 ) = −V0
"
x2
ρ( ~x1 +~
)
2
1−
ρc
!γ #
δ(~x1 − ~x2 )
(1)
is adopted. The parameters of this pairing force are adjusted in such a
way that the average pairing gap resulting from the HF-BCS calculation
(where the average is made by considering levels within ± 5 MeV from the
Fermi energy) agree with the result of the three-point formula (the so-called
(3)
∆odd (N + 1) formula [10]). The properties of the excited states are not
considered as input to fit the force parameters. For the tin isotopes, using
in the p-h channel the SIII force [11], we obtain that the agreement of the
average paring gap with the outcome of the three-point formula is within
∼100 keV in 120−126 Sn (if γ = 1) with ρc = 0.16 fm−3 and V0 = 640 MeV
fm3 . In fact, pairing is only included in the case of 120 Sn. For 132 Sn and
208
Pb, HF-BCS plus QRPA is replaced by HF plus RPA.
The HF-BCS equations are solved in coordinate space. The step and
upper limit for the radial coordinate are equal respectively to 0.1 fm and 25
fm. The continuum part of the single-particle spectrum has been discretized
by using box boundary conditions. Only the neutron single-particle states
1h11/2 , 2f7/2 , 1h9/2 , 1i13/2 , 3p3/2 are considered for the BCS equations. The
RPA or QRPA equations are solved in the configuration space using the same
technique as explained in [4]. We have found that the results are stable and
the energy-weighted sum rule (EWSR) is satisfied if an upper cutoff in the
two quasi-particle energies is set at 100 MeV.
In this self-consistent mean field model, the spurious states 0+ and 1−
associated respectively with the breaking in the superfluid ground state of
the number of particles, and with the translational symmetry, appear at
zero energy when the p-h matrix elements are renormalized by a few percent
(3%), because of the finite basis size. This little renormalization does not
affect significantly the dipole response at any energy. No renormalization of
the p-p interaction is needed for the 0+ state.
To include the coupling with the vibrations, we use the same Hamiltonian
of Ref. [12] in the full neutron-proton scheme. The formalism has been
described in detail in Ref. [4], to which we confer the reader (cf. in particular
Sec. 2 and Appendix A). In the present case the phonons of our model are
the states resulting from RPA or QRPA calculations of multipolarity 1− ,
2+ , 3− and 4+ having energy below 30 MeV and exhausting at least 5% of
4
isoscalar or isovector strength. The properties of the low-lying 2+ and 3−
states in the Sn isotopes are discussed in ref. [13].
We have checked by means of an explicit calculation for the 208 Pb case,
that continuum-RPA gives peak positions quite similar to our discrete RPA,
with escape widths smaller than 0.5 MeV, in agreement with the experimental
data [2].
We briefly discuss the sensitivity of the dipole results to the particular
Skyrme force employed, before analyzing in detail the results obtained using
the parametrization SIII. The issue has been studied in Ref. [14]. In that
work, it was found by exploting the 208 Pb case that the RPA centroid energy
E−1 (defined as (m1 /m−1 )1/2 , where mk is the k-th moment of the strength
function) of the dipole, has a linear dependence on the quantity FD associated
with the parameters of the Skyrme force in question,
E−1 = λ + µFD .
(2)
The explicit expression of FD can be found in Ref. [14], but we remind
here that it contains essentially the square root of the product of the bulk
symmetry energy aτ (sometimes indicated by J or a4 ), of the ratio between
the surface symmetry energy coefficient ass and aτ , and of the effective mass
m∗ . The dimensions of FD are MeV1/2 .
It is confortable to have found in the present context that the linear
relation (2) is valid also for 120 Sn. While in the 208 Pb case, the values λ=11.35
MeV and µ=0.77 MeV1/2 were obtained in [14], in the case of 120 Sn (132 Sn)
we find in the present analysis λ=9.28 MeV (11.92 MeV) and µ=2.34 MeV1/2
(1.46 MeV1/2 ). For 120 Sn, forces like SkP [15], SLy4 [16] and SGII [17] give
results for the peak energy which are lower than the experimental ones. This
is not the case for the force SIII, and we will use it, since the phonon coupling
induces a downward shift of the RPA peak energy.
To be more quantitative, in Table 1, we compare the mean field results
obtained with the SIII and SLy4 forces. We report the centroid energies
E0 = m1 /m0 obtained within RPA or QRPA, together with the unperturbed
HF or HF-BCS results, and with the phenomenological predictions. We can
observe that, for all nuclei we consider, the unperturbed value of the centroid
obtained using SIII is similar (yet slightly lower) than that obtained using
SLy4, whereas the RPA result is sistematically higher. In fact, while for the
two forces the effective mass is about the same (m∗ /m = 0.76 for SIII and
m∗ /m = 0.7 for SLy4), the repulsive matrix elements obtained with SIII are
5
nucleus
SIII
SLy4
80 A−1/3 (41 A−1/3 )
208
Pb
14.9 (9.4) 13.4 (9.8)
13.5 (6.9)
120
Sn (RPA) 17.1 (10.9) 15.0 (11.0)
16.2 (8.3)
120
Sn (QRPA) 17.3 (11.3) 15.1 (11.6)
16.2 (8.3)
132
Sn
17.0 (10.7) 15.2 (11.0)
15.7 (8.1)
m1
obtained within RPA or QRPA (together
Table 1: Centroid energies E0 = m
0
with the unperturbed HF or HF-BCS result in parenthesis), compared with
the empirical prediction 80 A−1/3 (41 A−1/3 ).
sistematically larger that those of the SLy4. From the comparison between
RPA and QRPA results for the nucleus 120 Sn, we can deduce that the pairing
interaction does not play a crucial role, as expected.
In Table 2, the results for the centroid energy without and with the
phonon coupling are shown. In the case of 208 Pb, the centroid energy after
the coupling is 14.4 MeV, to be compared with the value 14.9 MeV at the
level of RPA. In the case of 120 Sn, the centroid energy, at the level of the
complete calculation is the same value of the QRPA calculation. The same
is true for 132 Sn.
The integral of the strength as a function of the upper integration limit,
that is, the cumulated value of the EWSR, is shown in Fig. 1. This cumulated
value is shown as a fraction of the total expected value, which is the TRK
sum rule multiplied by the enhancement factor (1 + κ). The values of the
total EWSR in MeV·fm2 (and of κ in parenthesis) are, for 120 Sn, 132 Sn and
208
Pb respectively: 2448 (0.399), 2566 (0.397) and 4202 (0.408). One can see
that, in order to exhaust more than 90-95% of the EWSR in the (Q)RPA-PC
calculation, one has to reach 30 MeV in the case of 208 Pb and 40 MeV in
the case of the Sn isotopes. We stress that, while the shape of the strength
distribution and the GDR peak position change when the particle-vibration
coupling is taken in account, the centroid and the EWSR values are much
less affected.
In the Figures 2-4, we show the calculated photoabsorption cross sections,
together with the experimental results for the two stable nuclei. The values
of energies and the widths obtained from a lorentzian fit of our cross sections
are summarized in Table 3.
We can see that the agreement with the experimental results is good
6
for both 208 Pb and 120 Sn. This gives us confidence to use the model for
a prediction of the photoabsorbtion in the unstable nucleus 132 Sn. In this
case the width is about 6 MeV, like in 120 Sn, but the cross section is very
fragmented.
In the low-lying energy region, we obtain strength in the three nuclei,
as we discuss below. The cumulated value of the EWSR below 12 MeV is
shown in Fig. 5. For 208 Pb, a comparison is possible with the data of Ref. [9].
Our calculation gives a total integrated strength B(E1)=0.53 e2 fm2 (below 6
MeV) and B(E1)=1.27 e2 fm2 (up to 8 MeV), whereas the experimental data
are B(E1)=0.52 e2 fm2 (below 6 MeV) and B(E1)=0.80 e2 fm2 (up to 8 MeV).
The states carrying this strength are mainly single-particle excitations, so the
value of their energy is affected by the single-particle energy spectrum. On
the other hand, the global B(E1) distribution is well reproduced qualitatively
and quantitatively.
Coming to the Sn isotopes, the low-lying dipole strength grows faster,
at the level of (Q)RPA, in 120 Sn. The phonon coupling affects more 132 Sn,
reversing somewhat the trend. Also from Figs. 4 and 3 one can see that
in 132 Sn four definite peaks exhaust the strength below 10 MeV, whereas in
120
Sn only a kind of smooth background is visible.
We have analyzed the structure of the four lying below 10 MeV in 132 Sn.
In RPA, the lowest peak at 8.44 MeV is mainly (78%) associated with the
3s1/2 → 3p3/2 single-particle transition, and absorbs only 0.2% of the EWSR.
There are two higher peaks which lie at 8.61 MeV and 9.53 MeV and absorb
respectively 0.5% and 0.3% of the EWSR: they involve an admixture of the
2d3/2 → 3p1/2 and 3s1/2 → 3p1/2 transitions. Other small states involve the
d → f transitions but there is no peak to which more than two or three components contribute. The pattern is very similar if we move to the interaction
SLy4. This situation is somewhat at variance with the findings of Ref. [5],
in which the lowest state emerging from the relativistic RPA has four or five
configuration well mixed and it exhausts alone about 1.4% of the EWSR.
We conclude that an experiment aimed to explore the existence of either a
well defined, relatively collective peak or a fragmented pattern in the lowlying energy region can really discriminate among microscopic self-consistent
models.
The analysis of the RPA-PC wave functions of the states is slightly more
difficult and less transparent, also because of the use of a complex Hamiltonian. However, we can state that the wave functions do not acquire more
collectivity. This is confirmed by considering the transition densities, which
7
nucleus
208
Pb
120
Sn
132
Sn
RPA
14.9
17.1
17.0
QRPA-PC
14.4
17.1
17.0
1
Table 2: Centroid energies E0 = m
in MeV, obtained in the (Q)RPA and
m0
(Q)RPA-PC calculations. Only in the case of 208 Pb we observe an appreciable
shift when the phonon coupling is introduced. In the other two cases the
downward shift is smaller than 100 keV.
nucleus
208
Pb
120
Sn
132
Sn
QRPA-PC
exp.
13.1 (3.7) 13.46 (3.9)
15.7 (5.3) 15.4 (4.9)
15.5 (5.8)
-
Table 3: Values of the peak energy (width) in MeV calculated in (Q)RPA-PC
(calculated by means of a lorentzian fit to the cross sections) in comparison
with the experimental values.
are reported in Fig. 6 and are associated indeed to the RPA-PC calculation.
The transition density associated with the the GDR peak at 13.7 MeV is
compared with that of the low-lying state at 9.7 MeV. This latter has many
nodes while that of the GDR displays the expected shape of collective type.
The low-lying transition densities is also dominated by the neutron contribution at the surface and this fact is of course relevant for any consideration
concerning its excitation by means direct reactions. The neutron character
of the low-lying state at the surface is even more pronounced in the complete
calculation, than in simple RPA.
To judge the collectivity of the low-lying strength, one may think about
using the so-called “cluster sum rule” Sclus [18]. However, as stated in many
occasions, its use is jusitified only provided an unambigous decoupling of
excitations of core and valence particles is present in the physical systems.
Already from the above discussion, and in particular from the analysis of the
wave functions, it is clear that this is not the case for 132 Sn: all the excess
neutrons contribute to the low-lying strength and not only the least bound
ones.
8
In conclusion, we have implemented with the present work a complete
model which includes states of four quasi-particles nature in the description
of the vibrational nuclear states. The model makes self-consistent use of
a Skyrme force in the particle-hole channel and of a zero-range, densitydependent pairing force in the particle-particle channel. We have applied
this model to the description of the the dipole strength in different systems,
reaching satisfactory results. In 120 Sn and 208 Pb we obtained agreement with
the experimental data, for the peak energy and width of the GDR (and for the
pygmy states recently studied in 208 Pb). In 132 Sn, we have predictions which
show a large fragmentation of the main GDR and the absence of collective
states in the low-lying region. This latter result, obtained using two different
effective forces, is somehow at variance with the outcome of relativistic RPA
studies [5]. However, it is consistent with the discussion in Ref. [19], where it
is argued that the soft dipole strength, observed in halo light nuclei, should
decrease in skin nuclei, due to the coupling to the GDR. These remarks point
to the importance of clear-cut experimental data which are able to identify or
exclude the presence of a low-lying collective dipole in medum-heavy nuclei.
9
References
[1] P.F. Bortignon, A. Bracco and R.A. Broglia, Giant Resonances. Nuclear
Structure at Finite temperature (Harwood Ac. Publ., New York, 1998).
[2] M.N. Harakeh and A. van der Woude, Giant Resonances: Fundamental
High-Energy Modes of Nuclear Excitation (Oxford Un. Press, Oxford,
2001).
[3] A. Leistenschneider et al., Phys. Rev. Lett. 86 (2001) 5442.
[4] G. Colò and P.F. Bortignon, Nucl. Phys. A 696 (2001) 427.
[5] D. Vretenar, N. Paar, P. Ring and G.A. Lalazissis, Nucl. Phys. A 692
(2001) 496.
[6] Cf., e.g., the relevant papers in the Proc. of the Int. Symp. on Physics of
Unstable Nuclei, Nucl. Phys. A 722 (2003).
[7] G.F. Bertsch, P.F. Bortignon and R.A. Broglia, Rev. Mod. Phys. 55
(1983) 287.
[8] P.F. Bortignon and R.A. Broglia, Nucl Phys. A 371 (1981) 405.
[9] N. Ryezayeva et al., Phys. Rev. Lett. 89 (2002) 272502.
[10] W. Satula, J. Dobaczewski and W. Nazarewicz, Phys. Rev. Lett. 81
(1998) 3599; T. Duguet, P. Bonche, P.-H. Heenen, J. Meyer, Phys. Rev.
C 65 (2001) 014311.
[11] M. Beiner, H. Flocard, N. Van Giai and Ph. Quentin, Nucl. Phys. A 238
(1975) 29.
[12] G. Colò, T. Suzuki and H. Sagawa, Nucl. Phys. A 695 (2001) 167.
[13] G. Colò et al., Nucl. Phys. A 722 (2003) 111c.
[14] G. Colò, Nguyen Van Giai, H. Sagawa, Phys. Lett. B 363 (1995) 5.
[15] J. Dobaczewski, H. Flocard and J. Treiner, Nucl. Phys. A 422 (1984)
103.
10
[16] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, R. Schaeffer, Nucl. Phys.
A 635 (1998) 231.
[17] N. Van Giai and H. Sagawa, Phys. Lett. B 106 (1981) 379.
[18] Y. Alhassid, M. Gai and G.F. Bertsch, Phys. Rev. Lett. 49 (1982) 1482.
[19] H. Sagawa, H. Esbensen, Nucl. Phys. A 693 (2001) 448.
11
Fraction of EWSR
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
208
10
20
Pb
30
132
10
20
Sn
30
120
10
20
Energy [MeV]
Sn
30
Figure 1: Exhaustion of the dipole EWSR in the three nuclei considered in
the present paper. The full line refers to the complete (Q)RPA calculation,
while the dashed line to the (Q)RPA calculation.
12
208
Pb dipole
photoabsorbtion cross section SIII
1000
lorentzian (exp fit)
RPA
RPA−PC
800
σ [mb]
600
400
200
0
5
10
15
20
25
30
E [MeV]
Figure 2: Photoabsorption cross section for
120
208
Pb.
Sn dipole
photoabsorbtion cross section SIII
600
lorentzian (exp. fit)
QRPA
QRPA−PC
500
σ [mb]
400
300
200
100
0
5
10
15
20
25
30
E [MeV]
Figure 3: Photoabsorption cross section for
13
120
Sn.
132
Sn dipole
photoabsorbtion cross section SIII
400
RPA
RPA−PC
σ [mb]
300
200
100
0
5
10
15
20
25
30
E [MeV]
Figure 4: Photoabsorption cross section for
14
132
Sn.
Fraction of EWSR
0.1
0.08
0.06
0.04
0.02
0
8
0.1
0.08
0.06
0.04
0.02
0
8
0.1
0.08
0.06
0.04
0.02
0
8
208
9
10
Pb
11
12
132
9
10
Sn
11
12
120
9
10
Energy [MeV]
11
Sn
12
Figure 5: Low-lying dipole strength in the three isotopes studied. The figure
has the same pattern of Fig. 1, but it refers only to the interval 8–12 MeV.
15
0.06
0.1
Protons
Neutrons
0.04
IS
IV
0.05
0.02
0
0
-0.02
-0.05
-0.04
-0.06
-0.1
-0.08
-0.1
-0.15
0
2
4
6
0.1
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
8
r [fm]
10
12
14
0
2
4
6
8
r [fm]
10
12
14
12
14
0.2
Protons
Neutrons
IS
IV
0.15
0.1
0.05
0
-0.05
0
2
4
6
8
r [fm]
10
12
14
0
2
4
6
8
r [fm]
10
Figure 6: Transition densities in 132 Sn at the level of RPA-PC. The lower
panels correspond to the GDR at 13.5 MeV while the two upper panels to
the low-lying state at 9.7 MeV MeV. Proton/neutron (isoscalar/isovector)
transition densities are shown in the left (right) side.
16